Radar signal processing with forward-backward matrix

ABSTRACT

Aspects of the present disclosure are directed to radar signal processing apparatuses and methods. As may be implemented in accordance with one or more embodiments, digital signals representative of received reflections of radar signals transmitted towards a target are mathematically processed to provide or construct a matrix pencil based on or as a function of a forward-backward matrix. Eigenvalues of the matrix pencil are computed and an estimation of the direction of arrival (DoA) of the target is output based on the computed eigenvalues.

Aspects of various embodiments are directed to estimating a direction ofarrival (DoA) of radar signal reflections from a target.

A variety of radar communications may be utilized for many differentapplications. For instance, radar communications may utilizehigh-resolution imaging radar technology in which computational andalgorithmic enhancements are employed to achieve angular resolutionsuperior to the natural resolution provisioned by the physical apertureof an antenna array of the radar system. However, achieving suchresolution can be challenging. For instance, as radar targets may beilluminated by the same source, the received echoes can be highlycorrelated, resulting is an array signal covariance that may need to befurther decorrelated before it can be useful. Therefore, an additionaldecorrelation process such as so-called spatial smoothing may be needed.Spatial smoothing may also be needed for obtaining multiple snapshots ofarray measurements. However, spatial smoothing approaches may requirehigh computational cost.

These and other matters have presented challenges to radarimplementations, for a variety of applications.

SUMMARY

Various example embodiments are directed to issues such as thoseaddressed above and/or others which may become apparent from thefollowing disclosure concerning radar signal processing and relateddetermination of DoA of a target or targets.

In accordance with a particular embodiment, a method includesmathematically processing, via logic circuitry, digital signalsrepresentative of received reflections of radar signals transmittedtowards a target to provide or construct a matrix pencil based on or asa function of a forward-backward matrix. Eigenvalues of the matrixpencil are computed, and an estimation of the DoA of the target isoutput based on the computed eigenvalues of the matrix pencil.

Another embodiment is directed to an apparatus comprising communicationcircuitry to transmit radar signals and to receive reflections of theradar signals from a target, and processing circuitry processrepresentative signals. Specifically, the processing circuitrymathematically processes digital signals representative of the receivedreflections of the radar signals to provide or construct a matrix pencilbased on or as a function of a forward-backward matrix. The processingcircuitry further computes eigenvalues of the matrix pencil, and outputsan estimation of the DoA of the target based on the computed eigenvaluesof the matrix pencil.

The above discussion/summary is not intended to describe each embodimentor every implementation of the present disclosure. The figures anddetailed description that follow also exemplify various embodiments.

BRIEF DESCRIPTION OF FIGURES

Various example embodiments may be more completely understood inconsideration of the following detailed description in connection withthe accompanying drawings, in which:

FIG. 1 is a system-level diagram illustrating an example radarcommunications system/apparatus, in accordance with the presentdisclosure;

FIG. 2 shows notional multi-input, multi-output (MIMO) linear-chirp FMCWradar apparatus with forward-backward matrix DoA determination, inaccordance with the present disclosure;

FIG. 3 shows an array sampling process with matrix calculation, inaccordance with the present disclosure;

FIG. 4 shows Hankel matrix construction with horizontal concatenation,in accordance with the present disclosure;

FIG. 5 shows Hankel matrix construction with vertical concatenation, inaccordance with the present disclosure;

FIG. 6 shows Toeplitz matrix construction with horizontal concatenation,in accordance with the present disclosure; and

FIG. 7 shows Toeplitz matrix construction with vertical concatenation,in accordance with the present disclosure.

While various embodiments discussed herein are amenable to modificationsand alternative forms, aspects thereof have been shown by way of examplein the drawings and will be described in detail. It should beunderstood, however, that the intention is not to limit the disclosureto the particular embodiments described. On the contrary, the intentionis to cover all modifications, equivalents, and alternatives fallingwithin the scope of the disclosure including aspects defined in theclaims. In addition, the term “example” as used throughout thisapplication is only by way of illustration, and not limitation.

DETAILED DESCRIPTION

Aspects of the present disclosure are believed to be applicable to avariety of different types of apparatuses, systems and methods involvingprocessing radar signal reflections to ascertain characteristics of oneor more targets from which the reflections are received. In certainimplementations, aspects of the present disclosure have been shown to bebeneficial when used in the context of automotive or other vehicularradar applications in which the DoA of a target is estimated usinghigh-resolution radar technology, as may be implemented for autonomousdriving (AD) and higher-level advanced driver assistance systems (ADAS).In a more particular embodiment, a matrix pencil is constructed using orbased on a forward-backward matrix, and eigenvalues of the matrix pencilare utilized for estimating the DoA of a target. For instance, highcomputational efficiency may be achieved via the employment of a superHankel matrix. In some implementations, such a matrix may replace thecostly process of constructing and eigen-decomposing spatially smoothedsignal covariance matrices. While not necessarily so limited, variousaspects may be appreciated through the following discussion ofnon-limiting examples which use exemplary contexts.

Accordingly, in the following description various specific details areset forth to describe specific examples presented herein. It should beapparent to one skilled in the art, however, that one or more otherexamples and/or variations of these examples may be practiced withoutall the specific details given below. In other instances, well knownfeatures have not been described in detail so as not to obscure thedescription of the examples herein. For ease of illustration, the samereference numerals may be used in different diagrams to refer to thesame elements or additional instances of the same element. Also,although aspects and features may in some cases be described inindividual figures, it will be appreciated that features from one figureor embodiment can be combined with features of another figure orembodiment even though the combination is not explicitly shown orexplicitly described as a combination.

In accordance with a particular embodiment digital signals, which arerepresentative of received reflections of radar signals transmittedtowards a target, are processed to provide or construct a matrix pencilbased on or as a function of a forward-backward matrix. The reflectionsmay be received in a radar circuit that transmits the radar signals, andin some implementations, the digital signals are processed as part of aninput array measurement vector (e.g., which may utilize the transmittedsignals). Eigenvalues (e.g., eigenvalue phases) of the matrix pencil maybe computed, and an estimation of the direction of arrival (DoA) of thetarget may be generated/output based on the computed eigenvalues of thematrix pencil. For instance, such an output may be based on eigenvaluesof the matrix pencil, and eigenvalue phases corresponding to certain ofthe eigenvalues having a magnitude within a predefined range. In someimplementations, halved-degrees-of-freedom issues associated with thematrix pencil and/or the forward-backward matrix are computationallyresolved and used with outputting the DoA estimation. Such approachesmay be used to estimate the DoA with sufficiently high-resolutionimaging based on resolving a number of targets greater than two-thirdsthe size of the array.

The forward-backward matrix may be constructed in a variety of manners.For instance, the forward-backward matrix may include or refer to amatrix having multiple concatenated sub-matrices. Each sub-matrix may becharacterized as a diagonal-constant matrix having a diagonal directionconsistent with either: each ascending diagonal from left to right beingconstant (e.g., Hankel-like) or each descending diagonal from left toright being constant (e.g., Toeplitz-like). The forward-backward matrixmay include respective forward and backward matrices having the samenumber of rows and columns.

Mathematically processing as noted above may be carried out in a varietyof manners. For instance, multiple matrices may be formed in which atleast two are characterized as a diagonal-constant matrix having adiagonal direction consistent with either: each ascending diagonal fromleft to right being constant (e.g., Hankel-like), or each descendingdiagonal from left to right being constant. (e.g., Toeplitz-like). Thesemay be used to construct a forward-backward matrix as noted. As anotherexample, a forward matrix and a backward matrix respectively constructedof the digital signals may be concatenated, horizontally from left toright and constructing the matrix pencil from the matrices. The forwardand backward matrices constructed of the reflections may be concatenatedvertically from top to bottom, and the matrix pencil may be constructedfrom the matrices. In one embodiment, forward and backward Hankelmatrices may be generated in which values of rows and columns of theforward and backward matrices are chosen so the resulting matrix is asquare or has one more row than columns. In other embodiments, forwardand backward Hankel matrices may be generated in which values of rowsand columns of the forward and backward matrices are chosen so theresulting matrix is a wide matrix which has more columns than rows toachieve better signal to noise performance in exchange of fewer numberof targets to be estimated. In one embodiment, forward and backwardToeplitz matrices may be generated, in which values of the rows andcolumns of the forward and backward matrices are chosen so the resultingmatrix is a square or has one more column than rows. In otherembodiments, forward and backward Toeplitz matrices may be generated inwhich values of rows and columns of the forward and backward matricesare chosen so the resulting matrix is a tall matrix which has more rowsthan columns to achieve better signal to noise performance in exchangeof fewer number of targets to be estimated. Various such approaches mayalso be combined.

The steps of mathematically processing, computing and outputting mayomit, for example, providing or estimating a number of signal sourcesand eliminating noise eigenvectors in an eigenvector matrix oreigenvector matrices. These steps may omit constructing andEigen-decomposing spatially smoothed signal covariance matrices.Further, these steps may include providing the DoA within an angularresolution of less than 0.05 in a normalized frequency scale of 0-1.

Another embodiment is directed to an apparatus having communicationcircuitry and processing circuitry. The communication circuitrytransmits radar signals and receives reflections of the radar signalsfrom a target. The processing circuitry mathematically processes digitalsignals representative of the received reflections of the radar signals(e.g., as part of an input array measurement vector) to provide orconstruct a matrix pencil based on or as a function of aforward-backward matrix. The processing circuitry further computeseigenvalues of the matrix pencil, and outputs an estimation of the DoAof the target based on the computed eigenvalues.

The processing circuitry may operate in a variety of manners. In aparticular embodiment, the processing circuitry forms multiple matrices,at least two of which are characterized as a diagonal-constant matrixhaving a diagonal direction consistent with either: each ascendingdiagonal from left to right being constant, or each descending diagonalfrom left to right being constant.

Further, a variety of forward-backward matrices may be utilized. Forinstance, the forward-backward matrix may include or refer to a matrixhaving multiple concatenated sub-matrices, each of which ischaracterized as a diagonal-constant matrix having a diagonal directionconsistent with either: each ascending diagonal from left to right beingconstant, or each descending diagonal from left to right being constant.

Turning now to the figures, FIG. 1 shows a radar apparatus 100, as maybe implemented in accordance with one or more embodiments. The apparatus100 includes an antenna array 110, radar communication circuitry 120,and radar processing circuitry 130 (which may further interface withinterface circuitry 140, for example automotive interface circuitry).The antenna array 110 includes a plurality of antennas, and the radarcommunication circuitry 120 includes transmission circuitry 122 andreception circuitry 124 (e.g., a plurality of transmitters andreceivers). The radar processing circuitry 130 (e.g., radar MCPU)includes a controller module 132, as well as matrix construction andpencil-based DoA computation circuitry 134.

These components of apparatus 100 are operable to provide radarcommunications, in connection with signals communicated with the radarprocessing circuitry 130, utilizing time-frequency domain oversampling,and as may be implemented in accordance with one or more embodimentsherein. For instance, positional characteristics including DoA of atarget from which radar signals transmitted by the transmissioncircuitry 122 via the antenna array 110, and which are reflected fromthe target and received by the reception circuitry via the antennaarray, may be ascertained by constructing a matrix pencil based on or asa function of a forward-backward matrix as characterized herein.Eigenvalues of the matrix pencil are computed and utilized to estimatethe DoA. In certain embodiments, the transmission circuitry 122 andreception circuitry 124 are respectively implemented in accordance withthe transmitter and receiver circuitry as characterized in communicationcircuitry 220 in FIG. 2 .

FIG. 2 shows notional multi-input, multi-output (MIMO) linear-chirp FMCWradar apparatus 200 with forward-backward matrix DoA determination, asmay be utilized in accordance with one or more embodiments. Theapparatus 200 includes an antenna array 210, communication circuitry 220and radar processing circuitry 230. The communication circuitry 220includes a plurality of transmitters and receivers (e.g., threetransmitters and four receivers). The transmitters may include a chirpgenerator 221 operable to generate respective chirps, and may includeother transmission circuitry such as conditioning and amplifiercircuits, and operate in response to radar control circuitry within theradar processing circuitry 230.

The receivers may include amplifier, filters and other circuits asuseful for receiving radar signals. For instance, each receiver may mixa return radar reflection with a transmitted chirp and filter the resultto generate deramped IF (intermediate frequency) signals to be sampledby analog-to-digital converters (ADCs) and processed by a digital signalprocessing (DSP) unit to produce range and Doppler responses for eachreceive channel. The range-Doppler response maps of the receivers fromthe transmitted signals may be aggregated to form a complete MIMO arraymeasurement data cube of range-Doppler response maps of antenna elementsof a constructed MIMO virtual array. The range-Doppler responses may benon-coherently integrated and target detection may be attempted on theenergy-combined range-Doppler map. A detection algorithm, such as mayrelate to variants of a CFAR algorithm, may be used to identifyrange-Doppler cells in which targets may be present. For each detectioncell, the array measurement vector may then be extracted and processedfor identifying the incident angles of any target returns contained inthe cell.

Reflected radar signals received via the antenna array 210 andcommunication circuitry 220 are passed to the radar processing circuitry230. The received signals are processed accordingly, including targetDoA estimation at block 232 using a forward-backward matrix and matrixpencil as characterized herein (e.g., as with FIG. 1 ). The radarprocessing circuitry 230 may further carry out angle estimation andtarget tracking, using output array vectors, for tracking positionalcharacteristics of targets from which reflections are received. Suchtarget information may be provided via a data interface to externalsystems (e.g., automotive).

FIG. 3 shows an array sampling process with matrix calculation, inaccordance with the present disclosure. At block 301, an array samplevector corresponding to radar reflections is input and a left-right,top-bottom (LR/TB) super Hankel/Toeplitz matrix is constructed at block302. At block 303, a matrix pencil is computed directly from the LR/TBsuper Hankel/Toeplitz matrix, and generalized eigenvalues of the pencilare determined at block 304. Those eigenvalues with magnitudes close to1 (within a tolerance) are selected at block 305 and their phases arecomputed at block 306. At block 307, the DoA of a target from which theradar reflections are received is computed from the phase informationand DoA estimates of the target are output at block 308. Such anapproach may, for example, utilize computations as characterized aboveand/or with FIGS. 4-7 that follow.

FIG. 4 shows Hankel matrix construction with horizontal concatenation,in accordance with the present disclosure. At 400 an array vector s isused to generate a forward Hankel matrix 401, and a backwards arrayvector S at 410 is used to generate a backwards Hankel matrix 411. Aleft-right (LR) super Hankel matrix 420 is then formed by concatenatingthe forward and backward Hankel matrices 401 and 411 horizontally.

The forward and backward Hankel matrices may be constructedindependently from the array measurements. For instance, for a vector ofN elements,S=[s ₁ ,s ₂ , . . . ,s _(N)]^(T) or [s ₁ ,s ₂ , . . . ,s _(L) ,s _(L+1), . . . s _(L+M−1)]^(T),wherein N=L+M−1, two sub-vectors overlapped on the L-th element may beformed. These sub-vectors, [s₁, s₂, . . . , s_(L)]^(T) and [s_(L),s_(L+1), . . . s_(L+M−1)]^(T), span the first column and the last row ofthe forward Hankel matrix 401, respectively.

The backward Hankel matrix may be formed similarly, in which the vectoris index-reversed and complex conjugated before the construction.Specifically, a backward array measurement vector

=[

₁,

₂, . . .

_(N)]^(T)≡[s_(N)*, s_(N−1)*, . . . , s₁*]^(T) may be formed and twosub-vectors [

₁,

₂, . . . ,

_(L)]^(T) and [

_(L),

_(L+1),

_(L+M−1)]^(T) are constructed. A backward Hankel matrix 411 is thenconstructed by spanning the first column and the last row of theBackward Hankel matrix with the two sub-vectors respectively and settinganti-diagonal elements to be equal valued.

FIG. 5 shows Hankel matrix construction with vertical concatenation, inaccordance with the present disclosure. A top-bottom (TB) super Hankelmatrix 520 is formed by concatenating the forward and backward Hankelmatrices 401 and 411 vertically.

FIG. 6 shows Toeplitz matrix construction with horizontal concatenation,in accordance with the present disclosure in which a horizontallyconcatenated super Toeplitz matrix (LR super Toeplitz matrix) is formedby concatenating the forward and backward Hankel matrices. Array vectors (400) is used to generate a forward Toeplitz matrix 601, and backwardsarray vector S (410) is used to generate a backwards Toeplitz matrix611. For the vector 400 of N elements,s=[s ₁ ,s ₂ , . . . ,s _(N)]^(T) or [s ₁ ,s ₂ , . . . ,s _(L+1) , . . .s _(L+M−1)]^(T),where N=L+M−1, two sub-vectors [s_(L), s_(L−1), . . . , s₁]^(T) and[s_(L), s_(L)+1, . . . s_(L+M−1)]^(T), overlapped on the first element(s_(L)), may be formed for construction of a super Toeplitz matrix 620.These sub-vectors, [s_(L), s_(L−1), . . . , s₁]^(T) and [s_(L), s_(L+1),. . . s_(L+M−1)]^(T), span the first column and the first row of theforward Toeplitz matrix 601, respectively. Two index-reversed andcomplex conjugated sub-vectors [

_(L),

_(L−1), . . . ,

₁]^(T) and [

_(L),

_(L+1), . . .

_(L+M−1)]^(T) are constructed. The backward Toeplitz matrix 611 isconstructed by spanning the first column and the first row of thebackward Toeplitz matrix with the two sub-vectors respectively (e.g.,and diagonal elements set to be equal valued).

FIG. 7 shows Toeplitz matrix construction with vertical concatenation,in accordance with the present disclosure. A vertically concatenatedsuper Toeplitz matrix (TB super Toeplitz matrix) 720 is formed byconcatenating the forward and backward Toeplitz matrices.

Referring to FIGS. 4-7 , 2L targets are modelled in the TB superHankel/Toeplitz instead of 2M which translate to index from 1 to L inforward Hankel/Toeplitz matrix and index from L+1 to 2L in the backwardHankel/Toeplitz matrix. Accordingly, four different SuperHankel/Toeplitz matrices may thus be computed by:

-   -   1) concatenating the Forward and Backward Hankel matrices        horizontally to form the Left-Right (LR) Super Hankel matrix;    -   2) concatenating the Forward and Backward Hankel matrices        vertically to form the Top-Bottom (TB) Super Hankel matrix;    -   3) concatenating the Forward and Backward Toeplitz matrices        horizontally to form the Left-Right (LR) Super Toeplitz matrix;        and    -   4) concatenating the Forward and Backward Toeplitz matrices        horizontally to form the Top-Bottom (TB) Super Toeplitz matrix.        The sizes of the forward and backward Hankel submatrices may be        identical and share the dimensions (L rows and M columns). The        values of L and M may be chosen to so the final LR super Hankel        matrix/Toeplitz matrix, which is of the size of L rows by 2M        columns, is square or a one-row extra tall matrix (e.g., having        one more row than columns), which may be implemented to maximize        the number of targets detectable. Similarly, the values of L and        M may be chosen to so the TB super Hankel matrix/Toeplitz        matrix, which is of the size of 2L rows by M columns, is square        or a one-column extra wide matrix (e.g., having 1 more columns        than rows), as may be implemented to maximize the number of        targets detectable. Other sizes of the matrices can be utilized        for attaining higher signal to noise performance with fewer        targets to be estimated.

Once the construction of the LR/TB Super Hankel/Toeplitz matrix iscomplete. The matrix pencil pair may be directly formed from the superHankel/Toeplitz matrix and their general eigenvalues may be estimated.Final DoA estimates may be computed from the phase information of asubset of the eigenvalues whose magnitudes are closest to 1 within agiven tolerance.

Matrix pencils as referred to herein may be computed in a variety ofmanners. The following discussion exemplifies such manners, as may beimplemented in accordance with one or more embodiments. For instance,the following approach may be implemented with the method depicted inFIG. 3 , and/or with one or more of the matrix construction approachesdepicted in FIGS. 4-7 .

Beginning with a horizontally concatenated super Hankel/Toeplitz matrix,a Left-right (LR) super Hankel matrix is formed by concatenating theforward and backward Hankel matrices horizontally and can be written inthe following equation (2M targets modelled in the Super Hankel):

${\overset{\leftrightarrow}{H}}_{Signal} = {\begin{bmatrix}{\overset{\rightarrow}{H}}_{Signal} & {\overset{\leftarrow}{H}}_{Signal}\end{bmatrix} = \text{ }\begin{bmatrix}{\sum\limits_{i = 1}^{M}{\alpha_{i}z_{i}^{0}}} & {\sum\limits_{i = 1}^{M}{\alpha_{i}z_{i}^{1}}} & & {\sum\limits_{i = 1}^{M}{\alpha_{i}z_{i}^{M - 1}}} & {\sum\limits_{i = {M + 1}}^{2M}{\alpha_{i}^{*}z_{i}^{{- M} - L + 2}}} & {\sum\limits_{i = {M + 1}}^{2M}{\alpha_{i}^{*}z_{i}^{{- M} - L + 3}}} & {\sum\limits_{i = {M + 1}}^{2M}{\alpha_{i}z_{i}^{{- L} + 1}}} \\{\sum\limits_{i = 1}^{M}{\alpha_{i}z_{i}^{1}}} & {\sum\limits_{i = 1}^{M}{\alpha_{i}z_{i}^{2}}} & & {\sum\limits_{i = 1}^{M}{\alpha_{i}z_{i}^{M}}} & {\sum\limits_{i = {M + 1}}^{2M}{\alpha_{i}^{*}z_{i}^{{- M} - L + 3}}} & {\sum\limits_{i = {M + 1}}^{2M}{\alpha_{i}^{*}z_{i}^{{- M} - L + 4}}} & {\sum\limits_{i = {M + 1}}^{2M}{\alpha_{i}^{*}z_{i}^{{- L} + 2}}} \\{\sum\limits_{i = 1}^{M}{\alpha_{i}z_{i}^{2}}} & {\sum\limits_{i = 1}^{M}{\alpha_{i}z_{i}^{3}}} & \cdots & {\sum\limits_{i = 1}^{M}{\alpha_{i}z_{i}^{M + 1}}} & {\sum\limits_{i = {M + 1}}^{2M}{\alpha_{i}^{*}z_{i}^{{- M} - L + 4}}} & {\sum\limits_{i = {M + 1}}^{2M}{\alpha_{i}^{*}z_{i}^{{- M} - L + 5}}} & {\sum\limits_{i = {M + 1}}^{2M}{\alpha_{i}^{*}z_{i}^{{- L} + 2}}} \\ \vdots & \vdots & & \vdots & \vdots & \vdots & \vdots \\{\sum\limits_{i = 1}^{M}{\alpha_{i}z_{i}^{L - 2}}} & {\sum\limits_{i = 1}^{M}{\alpha_{i}z_{i}^{L - 1}}} & & {\sum\limits_{i = 1}^{M}{\alpha_{i}z_{i}^{M + L - 3}}} & {\sum\limits_{i = {M + 1}}^{2M}{\alpha_{i}^{*}z_{i}^{- M}}} & {\sum\limits_{i = {M + 1}}^{2M}{\alpha_{i}^{*}z_{i}^{{- M} + 1}}} & {\sum\limits_{i = {M + 1}}^{2M}{\alpha_{i}^{*}z_{i}^{- 1}}} \\{\sum\limits_{i = 1}^{M}{\alpha_{i}z_{i}^{L - 1}}} & {\sum\limits_{i = 1}^{M}{\alpha_{i}z_{i}^{L}}} & & {\sum\limits_{i = 1}^{M}{\alpha_{i}z_{i}^{M + L - 2}}} & {\sum\limits_{i = {M + 1}}^{2M}{\alpha_{i}^{*}z_{i}^{{- M} + 1}}} & {\sum\limits_{i = {M + 1}}^{2M}{\alpha_{i}^{*}z_{i}^{{- M} + 2}}} & {\sum\limits_{i = {M + 1}}^{2M}{\alpha_{i}^{*}z_{i}^{0}}}\end{bmatrix}_{L \times 2M}}$${\left( {L \geq {2M}} \right) = {\begin{bmatrix}z_{1}^{0} & \cdots & z_{M}^{0} & z_{M + 1}^{{- M} - L + 2} & \cdots & z_{2M}^{{- M} - L + 2} \\z_{1}^{1} & \cdots & z_{M}^{1} & z_{M + 1}^{{- M} - L + 3} & \cdots & z_{2M}^{{- M} - L + 3} \\ & \vdots & & & \vdots & \\z_{1}^{L - 1} & \cdots & z_{M}^{L - 1} & z_{M + 1}^{{- M} + 1} & \cdots & z_{2M}^{{- M} + 1}\end{bmatrix}\begin{bmatrix}\alpha_{1} & \cdots & 0 & & & \\ \vdots & \ddots & \vdots & & 0 & \\0 & \cdots & \alpha_{M} & & & \\ & & & \alpha_{M + 1}^{*} & \cdots & 0 \\ & 0 & & \vdots & \ddots & \vdots \\ & & & 0 & \cdots & \alpha_{2M}^{*}\end{bmatrix}}}\text{ }{\begin{bmatrix}z_{1}^{0} & z_{1}^{1} & & z_{1}^{M - 1} & & & & \\ \vdots & \vdots & \cdots & \vdots & & 0 & & \\z_{M}^{0} & z_{M}^{1} & & z_{M}^{M - 1} & & & & \\ & & & & z_{M + 1}^{0} & z_{M + 1}^{1} & & z_{M + 1}^{M - 1} \\ & 0 & & & \vdots & \vdots & \cdots & \vdots \\ & & & & z_{2M}^{0} & z_{2M}^{1} & & z_{2M}^{M - 1}\end{bmatrix} = {\overset{\leftrightarrow}{Z}\overset{\leftrightarrow}{F}\overset{\leftrightarrow}{Y}}}$

Likewise, the horizontally concatenated super Toeplitz matrix (LR superToeplitz Matrix) may be formed by concatenating the forward and backwardHankel matrices and can be written in the following equation. SuperToeplitz Matrix (2M targets modelled in the Super Toeplitz):

${{\overset{\leftrightarrow}{T}}_{Signal} = {\begin{bmatrix}{\overset{\rightarrow}{T}}_{Signal} & {\overset{\leftarrow}{T}}_{Signal}\end{bmatrix} = \text{ }{\begin{bmatrix}{\sum\limits_{i = 1}^{M}{\alpha_{i}z_{i}^{L - 1}}} & {\sum\limits_{i = 1}^{M}{\alpha_{i}z_{i}^{L}}} & & {\sum\limits_{i = 1}^{M}{\alpha_{i}z_{i}^{M + L - 2}}} & {\sum\limits_{i = {M + 1}}^{2M}{\alpha_{i}^{*}z_{i}^{{- M} + 1}}} & {\sum\limits_{i = {M + 1}}^{2M}{\alpha_{i}^{*}z_{i}^{{- M} + 2}}} & {\sum\limits_{i = {M + 1}}^{2M}{\alpha_{i}z_{i}^{0}}} \\{\sum\limits_{i = 1}^{M}{\alpha_{i}z_{i}^{L - 2}}} & {\sum\limits_{i = 1}^{M}{\alpha_{i}z_{i}^{L - 1}}} & & {\sum\limits_{i = 1}^{M}{\alpha_{i}z_{i}^{M + L - 3}}} & {\sum\limits_{i = {M + 1}}^{2M}{\alpha_{i}^{*}z_{i}^{- M}}} & {\sum\limits_{i = {M + 1}}^{2M}{\alpha_{i}^{*}z_{i}^{{- M} + 1}}} & {\sum\limits_{i = {M + 1}}^{2M}{\alpha_{i}^{*}z_{i}^{- 1}}} \\{\sum\limits_{i = 1}^{M}{\alpha_{i}z_{i}^{L - 3}}} & {\sum\limits_{i = 1}^{M}{\alpha_{i}z_{i}^{L - 2}}} & \cdots & {\sum\limits_{i = 1}^{M}{\alpha_{i}z_{i}^{M + L - 4}}} & {\sum\limits_{i = {M + 1}}^{2M}{\alpha_{i}^{*}z_{i}^{{- M} - 1}}} & {\sum\limits_{i = {M + 1}}^{2M}{\alpha_{i}^{*}z_{i}^{- M}}} & {\sum\limits_{i = {M + 1}}^{2M}{\alpha_{i}^{*}z_{i}^{- 2}}} \\ \vdots & \vdots & & \vdots & \vdots & \vdots & \vdots \\{\sum\limits_{i = 1}^{M}{\alpha_{i}z_{i}^{1}}} & {\sum\limits_{i = 1}^{M}{\alpha_{i}z_{i}^{2}}} & & {\sum\limits_{i = 1}^{M}{\alpha_{i}z_{i}^{M}}} & {\sum\limits_{i = {M + 1}}^{2M}{\alpha_{i}^{*}z_{i}^{{- M} - L + 3}}} & {\sum\limits_{i = {M + 1}}^{2M}{\alpha_{i}^{*}z_{i}^{{- M} - L + 4}}} & {\sum\limits_{i = {M + 1}}^{2M}{\alpha_{i}^{*}z_{i}^{{- L} + 2}}} \\{\sum\limits_{i = 1}^{M}{\alpha_{i}z_{i}^{0}}} & {\sum\limits_{i = 1}^{M}{\alpha_{i}z_{i}^{1}}} & & {\sum\limits_{i = 1}^{M}{\alpha_{i}z_{i}^{M - 1}}} & {\sum\limits_{i = {M + 1}}^{2M}{\alpha_{i}^{*}z_{i}^{{- M} - L + 2}}} & {\sum\limits_{i = {M + 1}}^{2M}{\alpha_{i}^{*}z_{i}^{{- M} - L + 3}}} & {\sum\limits_{i = {M + 1}}^{2M}{\alpha_{i}^{*}z_{i}^{{- L} + 1}}}\end{bmatrix}_{L \times 2M} = \text{ }{\begin{bmatrix}z_{1}^{L - 1} & \cdots & z_{M}^{L - 1} & z_{M + 1}^{{- M} + 1} & \cdots & z_{2M}^{{- M} + 1} \\z_{1}^{L - 2} & \cdots & z_{M}^{L - 2} & z_{M + 1}^{- M} & \cdots & z_{2M}^{- M} \\ & \vdots & & & \vdots & \\z_{1}^{0} & \cdots & z_{M}^{0} & z_{M + 1}^{{- M} - L + 2} & \cdots & z_{2M}^{{- M} - L + 2}\end{bmatrix}\begin{bmatrix}\alpha_{1} & \cdots & 0 & & & \\ \vdots & \ddots & \vdots & & 0 & \\0 & \cdots & \alpha_{M} & & & \\ & & & \alpha_{M + 1}^{*} & \cdots & 0 \\ & 0 & & \vdots & \ddots & \vdots \\ & & & 0 & \cdots & \alpha_{2M}^{*}\end{bmatrix}}}}}\text{ }{\begin{bmatrix}z_{1}^{0} & z_{1}^{1} & & z_{1}^{M - 1} & & & & \\ \vdots & \vdots & \cdots & \vdots & & 0 & & \\z_{M}^{0} & z_{M}^{1} & & z_{M}^{M - 1} & & & & \\ & & & & z_{M + 1}^{0} & z_{M + 1}^{1} & & z_{M + 1}^{M - 1} \\ & 0 & & & \vdots & \vdots & \cdots & \vdots \\ & & & & z_{2M}^{0} & z_{2M}^{1} & & z_{2M}^{M - 1}\end{bmatrix} = {\overset{\leftrightarrow}{U}\overset{\leftrightarrow}{F}\overset{\leftrightarrow}{Y}}}$

Where the number of sources is known, while creating the LR SuperHankel/Toeplitz Matrix one can make 2M=N_(s) (or N_(s)+1 for odd values)so the number of eigenvalues to be computed equals the number ofsources, which may improve the SNR as a stronger averaging will takeplace in the matrix pencil.

The left-right (LR) super Hankel/Toeplitz matrix may present arotational invariant property for use in the instant Matrix Pencilapproach. The

matrix in the LR super Hankel matrix model may be written in thefollowing form.

$\overset{\leftrightarrow}{Z} = \left\lbrack {\begin{matrix}Z_{1}^{0} & \ldots \\Z_{1}^{1} & \ldots \\ & {\vdots} \\Z_{1}^{L - 2} & \ldots \\Z_{1}^{L - 1} & \ldots\end{matrix}\begin{matrix}Z_{M}^{0} \\Z_{M}^{1} \\ \\Z_{M}^{L - 2} \\Z_{M}^{L - 1}\end{matrix}\begin{matrix}Z_{M + 1}^{{- M} - L + 2} \\Z_{M + 1}^{{- M} - L + 3} \\ \\Z_{M + 1}^{- M} \\Z_{M + 1}^{{- M} + 1}\end{matrix}\begin{matrix}\ldots \\\ldots \\ \vdots \\\ldots \\\ldots\end{matrix}\begin{matrix}Z_{2M}^{{- M} - L + 2} \\{Z_{2M}^{{- M} - L + 3}} \\Z_{2M}^{- M} \\Z_{2M}^{{- M} + 1}\end{matrix}} \right\rbrack$

A pair of submatrices (

_(A) and

_(B)) that are used to form the matrix pencil in the later step aremodelled by removing the last row of

to form

_(A), and by removing the first row of

to form

_(B), which can be written in the following form.

${\overset{\leftrightarrow}{Z}}_{A} = \begin{bmatrix}Z_{1}^{0} & \ldots & Z_{M}^{0} & Z_{M + 1}^{{- M} - L + 2} & \ldots & Z_{2M}^{{- M} - L + 2} \\Z_{1}^{1} & \ldots & Z_{M}^{1} & Z_{M + 1}^{{- M} - L + 3} & \ldots & Z_{2M}^{{- M} - L + 3} \\\begin{matrix} \\Z_{1}^{L - 2}\end{matrix} & \begin{matrix} \vdots \\\ldots\end{matrix} & \begin{matrix} \\Z_{M}^{L - 2}\end{matrix} & \begin{matrix} \\Z_{M + 1}^{- M}\end{matrix} & \begin{matrix} \vdots \\\ldots\end{matrix} & \begin{matrix} \\Z_{2M}^{- M}\end{matrix}\end{bmatrix}$ ${\overset{\leftrightarrow}{Z}}_{B} = \begin{bmatrix}Z_{1}^{1} & \ldots & Z_{M}^{1} & Z_{M + 1}^{{- M} - L + 3} & \ldots & Z_{2M}^{{- M} - L + 3} \\Z_{1}^{2} & \ldots & Z_{M}^{2} & Z_{M + 1}^{{- M} - L + 4} & \ldots & Z_{2M}^{{- M} - L + 4} \\\begin{matrix} \\Z_{1}^{L - 1}\end{matrix} & \begin{matrix} \vdots \\\ldots\end{matrix} & \begin{matrix} \\Z_{M}^{L - 1}\end{matrix} & \begin{matrix} \\Z_{M + 1}^{{- M} + 1}\end{matrix} & \begin{matrix} \vdots \\\ldots\end{matrix} & \begin{matrix} \\Z_{2M}^{{- M} + 1}\end{matrix}\end{bmatrix}$

_(B) can be factorized by

_(A) and a diagonal matrix, for instance as follows:

${\overset{\leftrightarrow}{Z}}_{B} = {\begin{bmatrix}Z_{1}^{1} & \ldots & Z_{M}^{1} & Z_{M + 1}^{{- M} - L + 3} & \ldots & Z_{2M}^{{- M} - L + 3} \\Z_{1}^{2} & \ldots & Z_{M}^{2} & Z_{M + 1}^{{- M} - L + 4} & \ldots & Z_{2M}^{{- M} - L + 4} \\\begin{matrix} \\Z_{1}^{L - 1}\end{matrix} & \begin{matrix} \vdots \\\ldots\end{matrix} & \begin{matrix} \\Z_{M}^{L - 1}\end{matrix} & \begin{matrix} \\Z_{M + 1}^{{- M} + 1}\end{matrix} & \begin{matrix} \vdots \\\ldots\end{matrix} & \begin{matrix} \\Z_{2M}^{{- M} + 1}\end{matrix}\end{bmatrix}\left\lbrack {\begin{matrix}Z_{1} & \\ & \ddots \\0 & \end{matrix}\begin{matrix}0 \\ \\Z_{2M}\end{matrix}} \right\rbrack}$${\overset{\leftrightarrow}{Z}}_{B} = {{{\overset{\leftrightarrow}{Z}}_{A}*{{diag}\left( \left\lbrack {Z_{1}\ldots Z_{2M}} \right\rbrack \right)}} = {{\overset{\leftrightarrow}{Z}}_{A}\overset{\leftrightarrow}{\Phi}}}$where $\overset{\leftrightarrow}{\Phi} = {\left\lbrack {\begin{matrix}Z_{1} & \\ & \ddots \\0 & \end{matrix}\begin{matrix}0 \\ \\Z_{2M}\end{matrix}} \right\rbrack \equiv \left\lbrack {\begin{matrix}e^{{- j}\kappa\rho_{1}} & \\ & \ddots \\0 & \end{matrix}\begin{matrix}0 \\ \\e^{{- j}\kappa\rho_{2M}}\end{matrix}} \right\rbrack}$which demonstrates the rotational or shift invariant property.

Similarly, for the LR Super Toeplitz matrix model. The

matrix in the LR super Toeplitz matrix model may be written in thefollowing form.

$\overset{\leftrightarrow}{U} = \left\lbrack {\begin{matrix}Z_{1}^{L - 1} & \ldots \\Z_{1}^{L - 2} & \ldots \\ & {\vdots} \\Z_{1}^{0} & \ldots\end{matrix}\begin{matrix}Z_{M}^{L - 1} \\Z_{M}^{L - 2} \\ \\Z_{M}^{0}\end{matrix}\begin{matrix}Z_{M + 1}^{{- M} + 1} \\Z_{M + 1}^{- M} \\ \\Z_{M + 1}^{{- M} - L + 2}\end{matrix}\begin{matrix}\ldots \\\ldots \\ \vdots \\\ldots\end{matrix}\begin{matrix}Z_{2M}^{{- M} + 1} \\Z_{2M}^{- M} \\ \\Z_{2M}^{{- M} - L + 2}\end{matrix}} \right\rbrack$

A pair of submatrices (

_(A) and

_(B)) that are used to form the matrix pencil in the later step may bemodelled by removing the first row of

to form

_(A), and by removing the last row of

to form

_(B), which can be written in the following form. Rotational Invariance:

${\overset{\leftrightarrow}{U}}_{A} = \left\lbrack {\begin{matrix}Z_{1}^{L - 2} & \ldots \\Z_{1}^{L - 3} & \ldots \\ & {\vdots} \\Z_{1}^{0} & \ldots\end{matrix}\begin{matrix}Z_{M}^{L - 2} \\Z_{M}^{L - 3} \\ \\Z_{M}^{0}\end{matrix}\begin{matrix}Z_{M + 1}^{- M} \\Z_{M + 1}^{{- M} - 1} \\ \\Z_{M + 1}^{{- M} - L + 2}\end{matrix}\begin{matrix}\ldots \\\ldots \\ \vdots \\\ldots\end{matrix}\begin{matrix}Z_{2M}^{- M} \\Z_{2M}^{{- M} - 1} \\ \\Z_{2M}^{{- M} - L + 2}\end{matrix}} \right\rbrack$${\overset{\leftrightarrow}{U}}_{B} = \left\lbrack {\begin{matrix}Z_{1}^{L - 1} & \ldots \\Z_{1}^{L - 2} & \ldots \\ & {\vdots} \\Z_{1}^{1} & \ldots\end{matrix}\begin{matrix}Z_{M}^{L - 1} \\Z_{M}^{L - 2} \\ \\Z_{M}^{1}\end{matrix}\begin{matrix}Z_{M + 1}^{{- M} + 1} \\Z_{M + 1}^{- M} \\ \\Z_{M + 1}^{{- M} - L + 3}\end{matrix}\begin{matrix}\ldots \\\ldots \\ \vdots \\\ldots\end{matrix}\begin{matrix}Z_{2M}^{{- M} + 1} \\Z_{2M}^{- M} \\ \\Z_{2M}^{{- M} - L + 3}\end{matrix}} \right\rbrack$

In addition,

_(B) can be factorized by

_(A) and a diagonal matrix, as follows:

${\overset{\leftrightarrow}{U}}_{B} = {\left\lbrack {\begin{matrix}Z_{1}^{L - 2} & \ldots \\Z_{1}^{L - 3} & \ldots \\ & {\vdots} \\Z_{1}^{0} & \ldots\end{matrix}\begin{matrix}Z_{M}^{L - 2} \\Z_{M}^{L - 3} \\ \\Z_{M}^{0}\end{matrix}\begin{matrix}Z_{M + 1}^{- M} \\Z_{M + 1}^{{- M} - 1} \\ \\Z_{M + 1}^{{- M} - L + 2}\end{matrix}\begin{matrix}\ldots \\\ldots \\ \vdots \\\ldots\end{matrix}\begin{matrix}Z_{2M}^{- M} \\Z_{2M}^{{- M} - 1} \\ \\Z_{2M}^{{- M} - L + 2}\end{matrix}} \right\rbrack\left\lbrack {\begin{matrix}Z_{1} & \\ & \ddots \\0 & \end{matrix}\begin{matrix}0 \\ \\Z_{2M}\end{matrix}} \right\rbrack}$${\overset{\leftrightarrow}{U}}_{B} = {{{\overset{\leftrightarrow}{U}}_{A}*{{diag}\left( \left\lbrack {Z_{1}\ldots Z_{2M}} \right\rbrack \right)}} = {{\overset{\leftrightarrow}{U}}_{A}\overset{\leftrightarrow}{\Phi}}}$$\overset{\leftrightarrow}{\Phi} = {\left\lbrack {\begin{matrix}Z_{1} & \\ & \ddots \\0 & \end{matrix}\begin{matrix}0 \\ \\Z_{2M}\end{matrix}} \right\rbrack \equiv \left\lbrack {\begin{matrix}e^{{- j}\kappa\rho_{1}} & \\ & \ddots \\0 & \end{matrix}\begin{matrix}0 \\ \\e^{{- j}\kappa\rho_{2M}}\end{matrix}} \right\rbrack}$

A super Matrix Pencil problem as utilized herein may be solved based ona LR super Hankel/Toeplitz matrix as follows. Having established aboverotational invariant property, i.e.,

_(B)=

_(A)

, and

_(B)=

_(A)

, Super matrix pencil pair [

₁,

₂] (or equivalently [

₁ ^(H)

₁,

₁ ^(H)

₂]) for LR Super Hankel and matrix pencil pair for LR Super Toeplitz [

₁,

₂] (or equivalently [

₁ ^(H)

₁,

T₁ ^(H)

₂]) and the associated solver can be formulated based on the following.The duality exists for Hankel˜Toeplitz,

˜

,

_(A)˜

_(A),

_(B)˜

_(B),

˜

,

₁˜

₁ and

₂˜

₂ which means the formula below may be utilized by replacing currentsymbols with dual symbols.

Given Super Hankel matrix

, by definition,

=

_(Signal)+

=

+

.

₁, which is

with the last row removed, is formed with high SNR,

₁≅

_(A)(

).

₂, which is

with the first row removed, is formed with high SNR,

₂≅

_(B)(

)=

_(A)

(

) where

=diag[e^(−jκρ) ¹ e^(−jκρ) ² . . . e^(−jκρ) ^(2M) ].

The matrix pencil [

₁ ^(H)

₁,

₁ ^(H)

₂] may be formed and its generalized eigenvalues [μ₁ μ₂ . . . μ_(2M)]may be solved via QZ decomposition (e.g.,

₁ ^(H)

₂ε=μ

₁ ^(H)

₁ε, where μ∈[μ₁ μ₂ . . . μ_(2M)] is the generalized eigen value and εthe corresponding generalized eigen vector so μ_(1 . . . 2M) are alsothe eigenvalues of (

₁ ^(H)

₁)⁻¹

₁ ^(H)

₂). Conceptually this may be equivalent to solving the generalizedeigenvalues of matrix pencil [

₁,

₂].

For any eigenvalue y of the matrix pencil [

₁ ^(H)

₁,

₁ ^(H)

₂], det{μ(

₁ ^(H)

₁)−

₁ ^(H)

₂}=0 can be satisfied, such that

${\left. \rightarrow{\det\left\{ {{\overset{\leftrightarrow}{H}}_{1}^{H}{{\overset{\leftrightarrow}{Z}}_{A}\left( {{\mu I} - \overset{\leftrightarrow}{\Phi}} \right)}\left( {F\overset{\leftrightarrow}{Y}} \right)} \right\}} \right. = {\left. 0\rightarrow{\det\left( {{\mu I} - \overset{\leftrightarrow}{\Phi}} \right)} \right. = {\left. 0\rightarrow{\prod_{i = 1}^{2M}\left( {\mu - e^{{- j}\kappa\rho_{i}}} \right)} \right. = {\left. 0\rightarrow e^{{- j}\kappa\rho_{i}} \right. = {\left. \mu_{i}\rightarrow\rho_{i} \right. = \frac{\arg\left( \mu_{i} \right)}{- \kappa}}}}}},{\left. {and}\rightarrow{\cos\theta_{i}\sin\varphi_{i}} \right. = {\frac{\arg\left( \mu_{i} \right)}{- \kappa}{may}{also}{be}{satisfied}}},{{with}{eigenvalues}{having}{unit}{magnitudes}}$

-   -   (e.g., 1).

Accordingly, θ and φ can be solved based on the N_(s) eigenvalues whosemagnitudes are closest to 1 (∵ they correspond to main diagonals of

). For ULA, where it may not possible to unambiguously resolve both theelevation and azimuth angles, an elevation angle may be assumed forevaluating the azimuth angle. At least N_(s) eigenvalues, e.g., min{L−1,2M}≥N_(s), may be used.

Source amplitudes may be the main diagonals of

≅(

^(H)

)⁻¹

^(H)

^(H)(

^(H))⁻¹, with

,

formed with μ_(i)'s (valid for square or tall matrix

). For non-square/tall matrix

case amplitudes may be found by forming a steering matrix (A) withcolumns of steering vector pointing to the solved angles and solving forthe unknown amplitude vector (α) based on the known linear relationship(Aα=s) (e.g., ŝ=(A^(H)A)⁻¹A^(H)Aα in the least-squares sense).Alternatively, amplitude can be found by DFT's of ŝ tuned to μ_(i)'s.

A vertically concatenated super Hankel/Toeplitz matrix may beimplemented in accordance with the following. A top-bottom (TB) SuperHankel Matrix may be formed by concatenating Forward and Backward Hankelmatrices vertically and can be written in the following equation. 2Ltargets is modelled in the TB Super Hankel/Toeplitz instead of 2M whichtranslate to an index from 1 to L in a forward Hankel/Toeplitz matrixand an index from L+1 to 2L in the backward Hankel/Toeplitz matrix.

${{\overset{\leftrightarrow}{H}}_{Signal} = {\left\lbrack \text{⁠}\begin{matrix}{\overset{\rightarrow}{H}}_{Signal} \\{\overset{\leftarrow}{H}}_{Signal}\end{matrix} \right\rbrack = \left\lbrack \text{⁠}\begin{matrix}z_{1}^{0} & \cdots & z_{L}^{0} & & & \\z_{1}^{1} & \cdots & z_{L}^{1} & & & \\ & \vdots & & & & \\z_{1}^{L - 1} & \cdots & z_{L}^{L - 1} & & & \\ & & & z_{L + 1}^{{- M} - L + 2} & \cdots & z_{2L}^{{- M} - L + 2} \\ & & & z_{L + 1}^{{- M} - L + 3} & \cdots & z_{2L}^{{- M} - L + 3} \\ & & & & \vdots & \\ & & & z_{L + 1}^{{- M} + 1} & \cdots & z_{2L}^{{- M} + 1}\end{matrix} \right\rbrack}}\text{ }{{\left\lbrack \text{⁠}\begin{matrix}\alpha_{1} & \cdots & 0 & & & \\ \vdots & \ddots & \vdots & & 0 & \\0 & \cdots & \alpha_{L} & & & \\ & & & \alpha_{L + 1}^{*} & \cdots & 0 \\ & 0 & & \vdots & \ddots & \vdots \\ & & & 0 & \cdots & \alpha_{2L}^{*}\end{matrix} \right\rbrack\text{⁠}\left\lbrack \text{⁠}\begin{matrix}z_{1}^{0} & z_{1}^{1} & & z_{1}^{M - 1} \\ \vdots & \vdots & \cdots & \vdots \\z_{L}^{0} & z_{L}^{1} & & z_{L}^{M - 1} \\z_{L + 1}^{0} & z_{L + 1}^{1} & & z_{L + 1}^{M - 1} \\ \vdots & \vdots & \cdots & \vdots \\z_{2L}^{0} & z_{2L}^{1} & & z_{2L}^{M - 1}\end{matrix} \right\rbrack}\text{⁠} = {\overset{\leftrightarrow}{O}\overset{\leftrightarrow}{F}\overset{\leftrightarrow}{P}}}$

Likewise, the vertically concatenated super Toeplitz matrix (TB superToeplitz matrix) may be formed by concatenating the Forward and BackwardHankel matrices and can be written in the following equation. A superToeplitz Matrix (2L targets modelled in the Super Toeplitz) may beimplemented as follows:

${{\overset{\leftrightarrow}{T}}_{Signal} = {\left\lbrack \text{⁠}\begin{matrix}{\overset{\rightarrow}{T}}_{Signal} \\{\overset{\leftarrow}{T}}_{Signal}\end{matrix} \right\rbrack = \left\lbrack \text{⁠}\begin{matrix}z_{1}^{L - 1} & \cdots & z_{L}^{L - 1} & & & \\z_{1}^{L - 2} & \cdots & z_{L}^{L - 2} & & & \\ & \vdots & & & & \\z_{1}^{0} & \cdots & z_{L}^{0} & & & \\ & & & z_{L + 1}^{{- M} + 1} & \cdots & z_{2L}^{{- M} + 1} \\ & & & z_{L + 1}^{- M} & \cdots & z_{2L}^{- M} \\ & & & & \vdots & \\ & & & z_{L + 1}^{{- M} - L + 2} & \cdots & z_{2L}^{{- M} - L + 2}\end{matrix} \right\rbrack}}\text{ }{{\left\lbrack \text{⁠}\begin{matrix}\alpha_{1} & \cdots & 0 & & & \\ \vdots & \ddots & \vdots & & 0 & \\0 & \cdots & \alpha_{L} & & & \\ & & & \alpha_{L + 1}^{*} & \cdots & 0 \\ & 0 & & \vdots & \ddots & \vdots \\ & & & 0 & \cdots & \alpha_{2L}^{*}\end{matrix} \right\rbrack\text{⁠}\left\lbrack \text{⁠}\begin{matrix}z_{1}^{0} & z_{1}^{1} & & z_{1}^{M - 1} \\ \vdots & \vdots & \cdots & \vdots \\z_{L}^{0} & z_{L}^{1} & & z_{L}^{M - 1} \\z_{L + 1}^{0} & z_{L + 1}^{1} & & z_{L + 1}^{M - 1} \\ \vdots & \vdots & \cdots & \vdots \\z_{2L}^{0} & z_{2L}^{1} & & z_{2L}^{M - 1}\end{matrix} \right\rbrack}\text{⁠} = {\overset{\leftrightarrow}{B}\overset{\leftrightarrow}{F}\overset{\leftrightarrow}{P}}}$

If the number of sources is known, while creating the TB SuperHankel/Toeplitz Matrix one can readily make 2L=N_(s) (or N_(s)+1 for oddvalues) so the number of eigenvalues to be computed equals to the numberof sources. This may improve the SNR as a stronger averaging will takeplace in the matrix pencil.

Super matrix pencil methods as based on TB super Hankel/Toeplitz may beuseful for the recovery the DoA's of targets, utilizing rotationalinvariant property. The

matrix in the LR Super Hankel matrix model may be written in thefollowing form:

$\overset{\leftrightarrow}{P} = \left\lbrack {\begin{matrix}Z_{1}^{0} & Z_{1}^{1} \\ \vdots & \vdots \\Z_{L}^{0} & Z_{L}^{1} \\Z_{L + 1}^{0} & Z_{L + 1}^{1} \\ \vdots & \vdots \\Z_{2L}^{0} & Z_{2L}^{1}\end{matrix}\begin{matrix} \\\ldots \\ \\ \\\ldots \\

\end{matrix}\begin{matrix}Z_{1}^{M - 1} \\ \vdots \\Z_{L}^{M - 1} \\Z_{L + 1}^{M - 1} \\ \vdots \\Z_{2L}^{M - 1}\end{matrix}} \right\rbrack$

A pair of submatrices (

_(A) and

_(B)) that are used to form the matrix pencil in the later step may bemodelled by removing the last column of P to form EA, and by removingthe first column of

to form

_(B), which can be written in the following form.

${\overset{\leftrightarrow}{P}}_{A} = \left\lbrack {\begin{matrix}Z_{1}^{0} & Z_{1}^{1} \\ \vdots & \vdots \\Z_{L}^{0} & Z_{L}^{1} \\Z_{L + 1}^{0} & Z_{L + 1}^{1} \\ \vdots & \vdots \\Z_{2L}^{0} & Z_{2L}^{1}\end{matrix}\begin{matrix}{\ldots} \\ \\ \\ \\\ldots \\

\end{matrix}\begin{matrix}Z_{1}^{M - 2} \\ \vdots \\Z_{L}^{M - 2} \\Z_{L + 1}^{M - 2} \\ \vdots \\Z_{2L}^{M - 2}\end{matrix}} \right\rbrack$${\overset{\leftrightarrow}{P}}_{B} = \left\lbrack {\begin{matrix}Z_{1}^{1} & Z_{1}^{1} \\ \vdots & \vdots \\Z_{L}^{1} & Z_{L}^{1} \\Z_{L + 1}^{1} & Z_{L + 1}^{1} \\ \vdots & \vdots \\Z_{2L}^{1} & Z_{2L}^{1}\end{matrix}\begin{matrix} \\\ldots \\ \\ \\\ldots \\

\end{matrix}\begin{matrix}Z_{1}^{M - 1} \\ \vdots \\Z_{L}^{M - 1} \\Z_{L + 1}^{M - 1} \\ \vdots \\Z_{2L}^{M - 1}\end{matrix}} \right\rbrack$

Accordingly,

_(B) can be factorized by

_(A) and a diagonal matrix, for instance in the following equation:

${\overset{\leftrightarrow}{P}}_{B} = {\left\lbrack {\begin{matrix}Z_{1} & \\ & \ddots \\0 & \end{matrix}\begin{matrix}0 \\ \\Z_{2L}\end{matrix}} \right\rbrack\left\lbrack {\begin{matrix}Z_{1}^{0} & Z_{1}^{1} \\ \vdots & \vdots \\Z_{L}^{0} & Z_{L}^{1} \\Z_{L + 1}^{0} & Z_{L + 1}^{1} \\ \vdots & \vdots \\Z_{2L}^{0} & Z_{2L}^{1}\end{matrix}\begin{matrix} \\\ldots \\ \\ \\\ldots \\

\end{matrix}\begin{matrix}Z_{1}^{M - 2} \\ \vdots \\Z_{L}^{M - 2} \\Z_{L + 1}^{M - 2} \\ \vdots \\Z_{2L}^{M - 2}\end{matrix}} \right\rbrack}$${\overset{\leftrightarrow}{P}}_{B} = {{{{diag}\left( \left\lbrack {Z_{1}\ldots Z_{2L}} \right\rbrack \right)}*{\overset{\leftrightarrow}{P}}_{A}} = {\overset{\leftrightarrow}{\Phi}{\overset{\leftrightarrow}{P}}_{A}}}$where $\overset{\leftrightarrow}{\Phi} = {\left\lbrack {\begin{matrix}Z_{1} & \\ & \ddots \\0 & \end{matrix}\begin{matrix}0 \\ \\Z_{2L}\end{matrix}} \right\rbrack \equiv \left\lbrack {\begin{matrix}e^{{- {jk}}\rho_{1}} & \\ & \ddots \\0 & \end{matrix}\begin{matrix}0 \\ \\e^{{- {jk}}\rho_{2L}}\end{matrix}} \right\rbrack}$which demonstrates a rotational or shift invariant property.

In some embodiments, the super matrix pencil problem is solved based ona top-bottom (TB) super Hankel/Toeplitz matrix as follows. Referring tothe above rotational invariant property,

_(B)=

_(A), the super matrix pencil pair [

₁,

₂] (or equivalently [

₁

₁ ^(H),

₂

₁ ^(H)]) for TB super Hankel and matrix pencil pair for TB superToeplitz [

₁,

₂] (or equivalently [

₁

₁ ^(H),

₂

₁ ^(H)]) and the associated solver can be formulated based on thefollowing. The duality exists for Hankel˜Toeplitz,

˜

,

˜

,

₁˜

₁ and

₂˜

₂ which means the formula below may be used with current symbolsreplaced by dual symbols. Given Super Hankel matrix

, by definition,

=

_(Signal)+

=

+

.

₁, which is

with the last column removed, is formed with high SNR,

₁≅(

)

_(A).

₂, which is

with the first column removed, is formed. By definition with high SNR,

₂≅(

)

_(B)=(

)

_(A) where

=diag[e^(−jκρ) ¹ e^(−jκρ) ² . . . e^(−jκρ) ^(2L) ]

The matrix pencil [

₁

₁ ^(H),

₂

₁ ^(H)] may be formed and its generalized eigenvalues [μ₁ μ₂ . . .μ_(2M)] solved via QZ decomposition (e.g.,

₂

₁ ^(H)ε=μ

₁

₁ ^(H)ε, where με[μ₁ μ₂ . . . μ_(2M)] is the generalized eigen value andE the corresponding generalized eigen vector so μ_(1 . . . 2M) are alsothe eigenvalues of (

₁

₁ ^(H))⁻¹

₂

₁ ^(H)). Conceptually this may be equivalent to solving the generalizedeigenvalues of matrix pencil [

₁,

₂]. For instance, for any eigenvalue μ of the matrix pencil [

₁

₁ ^(H),

₂

₁ ^(H)], det{μ(

₁

₁ ^(H))−

₂

₁ ^(H)}=0 may be satisfied, such that

$\left. \rightarrow{\det\left\{ {\overset{\leftrightarrow}{O}{\overset{\leftrightarrow}{F}\left( {{\mu I} - \overset{\leftrightarrow}{\Phi}} \right)}{\overset{\leftrightarrow}{P}}_{A}{\overset{\leftrightarrow}{H}}_{1}^{H}} \right\}} \right. = {\left. 0\rightarrow{\det\left( {{\mu I} - \overset{\leftrightarrow}{\Phi}} \right)} \right. = {\left. 0\rightarrow{\prod_{i = 1}^{2M}\left( {\mu - e^{{- j}k\rho_{i}}} \right)} \right. = {\left. 0\rightarrow e^{{- j}k{\rho i}} \right. = {\left. \mu_{i}\rightarrow\rho_{i} \right. = {\left. \frac{\arg\left( \mu_{i} \right)}{- \kappa}\rightarrow{\cos\theta_{i}\sin\varphi_{i}} \right. = \frac{\arg\left( \mu_{i} \right)}{- \kappa}}}}}}$May also be satisfied with correct eigenvalues should have unitmagnitudes (e.g., 1).

Accordingly, θ and φ can be solved based on the N_(s) eigenvalues whosemagnitudes are closest to 1 (∵ they correspond to main diagonals of

) For ULA, it may not be possible to unambiguously resolve both theelevation and azimuth angles. Usually an elevation angle is assumed forevaluating the azimuth angle. At least N_(s) eigenvalues, e.g., min{2L,M−1}≥N_(s), may be used.

Source amplitudes may be the main diagonals of

≅(

^(H)

)⁻¹

^(H)

^(H)(

^(H))⁻¹ with

,

formed with μ_(i)'s (valid for square or fat matrix

). For a non-square/fat matrix

case, amplitudes may be found by forming a steering matrix (A) withcolumns of steering vector pointing to the solved angles and solving forthe unknown amplitude vector (α) based on the known linear relationship(Aα=s) (e.g. ŝ=(A^(H)A)⁻¹A^(H)Aα in the least-squares sense).Alternatively, amplitude can be found by DFT's of ŝ tuned to μ_(i)'s.

As examples, the Specification describes and/or illustrates aspectsuseful for implementing the claimed disclosure by way of variouscircuits or circuitry which may be illustrated as or using terms such asblocks, modules, device, system, unit, controller, interface circuitry,MCPU, and/or other circuit-type depictions (e.g., reference numerals 120and 130 of FIGS. 1, and 220 and 230 of FIG. 2 may depict a block/moduleas described herein). Such circuits or circuitry may be used togetherwith other elements to exemplify how certain embodiments may be carriedout in the form or structures, steps, functions, operations, activities,etc. As examples, wherein such circuits or circuitry may correspond tologic circuitry (which may refer to or include acode-programmed/configured CPU or MCPU), in one example the logiccircuitry may carry out a process or method (sometimes “algorithm”) byperforming sampling, transformation, oversampling and inversetransformation. In another example, logic circuitry may carry out aprocess or method by performing these same activities/operations and,various other radar processing steps in addition. Further operations,processes or methods in this context would be recognized in connectionwith the functions/activities associated with the processes depicted inFIGS. 3-7 .

For example, in certain of the above-discussed embodiments, one or moremodules are discrete logic circuits or programmable logic circuitsconfigured and arranged for implementing these operations/activities, asmay be carried out in the approaches shown in FIGS. 3-7 . In certainembodiments, such a programmable circuit is one or more computercircuits, including memory circuitry for storing and accessing a programto be executed as a set (or sets) of instructions (and/or to be used asconfiguration data to define how the programmable circuit is toperform). An algorithm or process as described in FIG. 3 and/or with thematrix computation(s) in one or more of FIGS. 4-7 , or as otherwisecharacterized herein for activities such as computing a target's DoA maybe used by the programmable circuit to perform the related steps,functions, operations, activities, etc. Depending on the application,the instructions (and/or configuration data) can be configured forimplementation in logic circuitry, with the instructions (whethercharacterized in the form of object code, firmware or software) storedin and accessible from a memory (circuit).

Based upon the above discussion and illustrations, those skilled in theart will readily recognize that various modifications and changes may bemade to the various embodiments without strictly following the exemplaryembodiments and applications illustrated and described herein. Forexample, methods as exemplified in the Figures may involve steps carriedout in various orders, with one or more aspects of the embodimentsherein retained, or may involve fewer or more steps. For instance, someembodiments are directed to fewer than all steps and/or components, suchas to carry out one or more types of matrix generation and subsequentuse thereof. Such modifications do not depart from the true spirit andscope of various aspects of the disclosure, including aspects set forthin the claims.

What is claimed is:
 1. A method comprising: processing in a radarsystem, the processing including: mathematically processing digitalsignals representative of received reflections of radar signalstransmitted towards a target to provide or construct a matrix pencilbased on or as a function of a forward-backward matrix; computingeigenvalues of the matrix pencil; and outputting an estimation of adirection of arrival (DoA) of the target based on the computedeigenvalues of the matrix pencil; and transmitting, in a communicationcircuit of the radar system, the radar signals and receiving thereflections of the transmitted radar signals; wherein mathematicallyprocessing includes processing the digital signals as part of an inputarray measurement vector, and, after computing the eigenvalues of thematrix pencil, selecting the computed eigenvalues of the matrix pencilwith magnitudes within a predefined range surrounding a magnitude of 1to use for the outputting of the estimation of the DoA of the target. 2.The method of claim 1, wherein mathematically processing includesforming multiple matrices, at least two of which are characterized as adiagonal-constant matrix having a diagonal direction consistent witheither: each ascending diagonal from left to right being constant, oreach descending diagonal from left to right being constant.
 3. Themethod of claim 1, wherein the forward-backward matrix includes orrefers to a matrix having multiple concatenated sub-matrices, each ofwhich is characterized as a diagonal-constant matrix having a diagonaldirection consistent with either: each ascending diagonal from left toright being constant or each descending diagonal from left to rightbeing constant.
 4. The method of claim 1, wherein the steps ofmathematically processing, computing, and outputting an estimation ofthe DoA use high-resolution imaging based on resolving a number oftargets greater than two-thirds of a number of rows or a number ofcolumns of the matrix.
 5. The method of claim 1, wherein the steps ofmathematically processing, computing, and outputting do not includeproviding or estimating a number of signal sources and eliminating noiseeigenvectors in an eigenvector matrix or eigenvector matrices.
 6. Themethod of claim 1, further comprising computing eigenvalue phasescorresponding to the computed eigenvalues having a magnitude within apredefined range, and wherein said outputting an estimation of the DoAis in response to computing the eigenvalue phases.
 7. The method ofclaim 1, wherein the steps of mathematically processing, computing, andoutputting do not include constructing and Eigen-decomposing spatiallysmoothed signal covariance matrices.
 8. The method of claim 1, whereincomputing eigenvalues of the matrix pencil includes computing eigenvaluephases of the matrix pencil.
 9. The method of claim 1, wherein the stepsof mathematically processing, computing, and outputting includeproviding the DoA within an angular resolution of less than 0.05 in anormalized frequency scale of 0-1.
 10. The method of claim 1, whereinoutputting an estimation of the DoA of the target based on eigenvaluephases of the matrix pencil is based on or a function of: eigenvalues ofthe matrix pencil; and eigenvalue phases corresponding to certain of theeigenvalues having a magnitude within a predefined range.
 11. The methodof claim 1, wherein mathematically processing the digital signalsincludes: concatenating a forward matrix and a backward matrixrespectively constructed of the digital signals, horizontally from leftto right; and constructing the matrix pencil from the matrices.
 12. Themethod of claim 1, wherein mathematically processing the digital signalsincludes: concatenating a forward matrix and a backward matrixrespectively constructed of the reflections, vertically from top tobottom; and constructing the matrix pencil from the matrices.
 13. Themethod of claim 1, wherein mathematically processing the digital signalsincludes generating forward and backward Hankel matrices in which valuesof rows and columns of the forward and backward matrices are chosen sothe resulting matrix is a square or has one more row than columns. 14.The method of claim 1, wherein mathematically processing the digitalsignals includes generating forward and backward Toeplitz matrices inwhich values of the rows and columns of the forward and backwardmatrices are chosen so the resulting matrix is a square or has one morecolumn than rows.
 15. The method of claim 1, wherein theforward-backward matrix includes forward and backward matrices havingthe same number of rows and columns.
 16. An apparatus comprising:communication circuitry to transmit radar signals and to receivereflections of the radar signals from a target; and processing circuitryto: mathematically process digital signals representative of thereceived reflections of the radar signals to provide or construct amatrix pencil based on or as a function of a forward-backward matrix;compute eigenvalues of the matrix pencil; and output an estimation of adirection of arrival (DoA) of the target based on the computedeigenvalues of the matrix pencil, wherein the processing circuitry is tofurther select the computed eigenvalues of the matrix pencil withmagnitudes within a predefined range surrounding a magnitude of 1 to usefor the outputting of the estimation of the DoA of the target.
 17. Theapparatus of claim 16, wherein the processing circuitry is to processthe digital signals as part of an input array measurement vector. 18.The apparatus of claim 16, wherein the processing circuitry is to formmultiple matrices, at least two of which are characterized as adiagonal-constant matrix having a diagonal direction consistent witheither: each ascending diagonal from left to right being constant, oreach descending diagonal from left to right being constant.
 19. Theapparatus of claim 16, wherein the forward-backward matrix includes orrefers to a matrix having multiple concatenated sub-matrices, each ofwhich is characterized as a diagonal-constant matrix having a diagonaldirection consistent with either: each ascending diagonal from left toright being constant, or each descending diagonal from left to rightbeing constant.